Properties

Label 2-1470-1.1-c1-0-3
Degree $2$
Conductor $1470$
Sign $1$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 6.24·11-s + 12-s + 5.65·13-s − 15-s + 16-s − 5.41·17-s − 18-s + 1.17·19-s − 20-s + 6.24·22-s + 8.82·23-s − 24-s + 25-s − 5.65·26-s + 27-s + 5.41·29-s + 30-s + 1.75·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.88·11-s + 0.288·12-s + 1.56·13-s − 0.258·15-s + 0.250·16-s − 1.31·17-s − 0.235·18-s + 0.268·19-s − 0.223·20-s + 1.33·22-s + 1.84·23-s − 0.204·24-s + 0.200·25-s − 1.10·26-s + 0.192·27-s + 1.00·29-s + 0.182·30-s + 0.315·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.292201301\)
\(L(\frac12)\) \(\approx\) \(1.292201301\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 6.24T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 + 5.41T + 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 - 8.82T + 23T^{2} \)
29 \( 1 - 5.41T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 - 6.48T + 41T^{2} \)
43 \( 1 + 5.07T + 43T^{2} \)
47 \( 1 + 6.24T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 - 8.82T + 59T^{2} \)
61 \( 1 - 0.343T + 61T^{2} \)
67 \( 1 + 5.07T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + 4.82T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.359389671984140178074224848775, −8.449848927606038986166160857196, −8.255652044158678091899951984206, −7.26040578021397072339642465224, −6.52779371903591659927446663777, −5.37612702172934133440579256508, −4.36440397428401516971861550638, −3.14514599817664623456611992581, −2.43280695826534350118871297251, −0.881882603000887135653074500524, 0.881882603000887135653074500524, 2.43280695826534350118871297251, 3.14514599817664623456611992581, 4.36440397428401516971861550638, 5.37612702172934133440579256508, 6.52779371903591659927446663777, 7.26040578021397072339642465224, 8.255652044158678091899951984206, 8.449848927606038986166160857196, 9.359389671984140178074224848775

Graph of the $Z$-function along the critical line